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Calculus and Optimization for Machine Learning
Week 6
Practice Quiz #2

Practice QuizPractice Quiz • 45 min45 minutes

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Practice Quiz #2

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Practice Quiz #2

Total points 5

1.

Question 1

For the function $f(x,y)=x^2-2xy+4y^3$ find all stationary point. As an answer, please, provide the product of its coordinates' means. If there is no stationary points. write $-1000$ .

1 / 1 point

0

0

Correct

Yes, you have found the local minimum at $(1/6; 1/6)$ and non extremum at $(0;0)$ !

2.

Question 2

Find the number of extrema of the function $f(x,y)=(x-1)^2+y^2-4y$ . If there is no extrema, write $-1$ .

1 / 1 point

1

1

Correct

Correct!

3.

Question 3

Choose correct statements (there could be more than one answer). In case of only positive arguments $x,y>0$ , the function $f(x,y)=Cx^{\alpha+1} y^{\beta+1}$ , $C,\alpha, \beta \ge 0$ , $\alpha+\beta<1$ :

1 / 1 point

has no stationary points;

Correct

Correct!

is convex or concave in neighborhoods of different points $(x,y)$ ;

is always concave;

has a stationary point at $(0;0)$;

is always convex;

Correct

Correct!

4.

Question 4

Assume the function $f(x,y)=x/y^2$ is given with a condition (restriction) $y-x+1=0$ . Amongst all the points that statisfy the condition, find extremal ones and write the $x$ coordinate of the minimum.

1 / 1 point

-1

-1

Correct

Correct!

5.

Question 5

Consider the function $(x^2+y^2) e^{-x^2-y^2}$ . Provide the number of maxima of $f(x,y)$ . In case there is no maxima, write $-1$ ; in case there is infinite number of maxima, write $-2$ .

1 / 1 point

-2

-2

Correct

Correct!